Prove that:
The family of $S$ of univalent functions on the unit disc with f(0)=0, f'(0)=1 is a normal family.
I'm pretty sure i have to do it with Zalcmans Lemma:
a family of analytic functions on the unit disc with f(0)=0 are not normal if and only if there exists:
(1) a sequence $(z_n)$ converging to $z_\infty$
(2)a sequence of positive reals $(p_n)$ convering to zero
(3) a sequence of functions in the family such that $f_n(z_n+p_nx)-f_n(z_n)$ converge on compact subsets to the entire identity function.
I know there are more complicated ways to treat the problem, but it is not supposed to be solved with advanced theory like the growth theorem/distortion theorem.
Please be so kind not to post a solution right away, but instead hint me how to solve the problem. Thank you very much for your time.
Best Answer
Try applying Montel's theorem to the related family of functions $\sqrt{f'(z)}$. Notice that the square roots may be chosen so that they all omit $-1$.